Probability methods

Practice

Recall it first, then use it — testing yourself builds the memory, where rereading the notes above only feels easier.

Recall and application shuffled together — you have to work out what each item wants before you can answer it, which is the part the exam actually tests. Both kinds feed the same spacing schedule, so anything you miss comes back sooner.

Say the answer out loud before flipping — that attempt is where the learning happens, even when you get it wrong.

Exam-style only. Instant feedback, shuffled each attempt, best score saved on this device.

What actually comes up

Three questions every year: a normal-distribution question (z-scores, inverse normal, and often a "why might normal not be appropriate" reasoning part), a probability tree built from a real scenario, and a two-way table with a relative-risk or expected-value comparison. The normal-distribution question has appeared every single year transcribed here, even though it can look like a separate topic from tree diagrams.

Across 3 years

The pattern — what to prepare

The 2025 paper ↗ Marking schedule ↗

Question One — Normal distribution and wait times

A cafe's muffins have mean weight 118 g, standard deviation 19 g; customer wait times on Saturday and Sunday are shown as frequency graphs.

• Muffins: μ = 118 g, σ = 19 g
• 10% of muffins below the minimum sellable weight

• a(i) Find the probability that a randomly chosen muffin weighs more than 100 g. Achieved
• a(ii) What percentage of muffins weigh between 100 g and 120 g? Achieved
• a(iii) 10% of muffins are below the minimum weight requirement. Find the maximum weight of a muffin that cannot be sold. Merit
• b Calculate a new standard deviation so no more than 2% of muffins weigh less than 100 g, keeping the mean at 118 g. Merit
• c Use the wait-time graphs to test a manager's claim, then compare and contrast the Saturday and Sunday distributions by shape, centre, and spread. Excellence

Question Two — Breakfast choices and a loyalty draw

Customers choose muesli/pancakes/bacon & eggs then fruit or yoghurt; a loyalty promotion has tiered prize probabilities.

• 25% muesli, 30% pancakes, rest bacon & eggs; 2/5 prefer fruit with muesli, 18% prefer yoghurt with pancakes
• 35% of customers spend over
  00; 1/5 of those win a prize; 40% of winners get a voucher

• a(i) Draw a probability tree for the breakfast/topping choices. What is the probability a customer chooses pancakes and fruit? Achieved
• a(ii) For 250 customers, how many would you expect to choose yoghurt? Merit
• b(i) Draw a probability tree for the loyalty promotion. What proportion of customers do not win? Achieved
• b(ii) What is the probability a customer who spends over
  00 wins a free coffee? Merit
• b(iii) Given the probability of winning a badge is 0.21, find the probability of winning a sticker, given a consolation prize. Excellence

Question Three — Coffee choices and anxiety

A two-way table of coffee choices by teenagers/adults; a claim about mocha preference; a table linking coffee consumption to anxiety.

• Coffee table: Mocha 76/150, Latte 184/115, Cappuccino 20/105 (teens/adults)
• Anxiety table: teenagers 267/390 total, adults total 210, grand total 600

• a From the coffee table, find the percentage ordering cappuccino, the probability a teenager orders a latte, and the probability of being an adult or a cappuccino drinker. Achieved
• b The owner believes adults are twice as likely as teenagers to order a mocha. Explain the owner's reasoning, then show with calculations that he is wrong. Merit
• c Complete the anxiety table, then evaluate the claim that teens are 80% more likely to be affected by coffee-induced anxiety than adults, using relative risk. Excellence

The 2024 paper ↗ Marking schedule ↗

Question One — Normal distribution and physical activity

Young people (5-17) do a mean 10.8 hours/week of physical activity, σ = 3.2; Year 11-12 students have a different mean and low-end percentile; elderly (75+) data is skewed.

• Young people: μ = 10.8 hrs, σ = 3.2 hrs
• Students 15-17: μ = 8.1 hrs; 5% spend under 0.5 hrs
• Elderly 75+: 45% spend under 0.5 hrs, μ = 4.3 hrs; excluding zero-activity, μ = 7.2 hrs

• a(i) Find the probability a randomly chosen young person spent between 6 and 10.8 hours of physical activity. Achieved
• a(ii) Find the probability a randomly chosen young person spent over 16 hours. Achieved
• a(iii) 12% of young people do less than the guideline amount. Find the minimum weekly hours needed to meet the guideline. Merit
• b(i) Given 5% of 15-17 year olds spend under 0.5 hours, and mean 8.1 hours, calculate the standard deviation. Merit
• b(ii) Give TWO reasons why this standard deviation suggests a normal distribution may not be appropriate. Excellence
• c For elderly people (45% under 0.5 hrs, mean 4.3 hrs, or 7.2 hrs excluding zero-activity), estimate the median and compare centre, spread, and shape to a normal distribution. Excellence

Question Two — A language-learning app

App users split by age (under 18 / 18-30 / over 30) and purpose (academic/travel/other); course-completion rates vary by purpose and by premium vs standard version.

• 22% under 18, 35% aged 18-30, rest over 30
• Under 18: 3/4 academic, 15% travel, rest other
• 18-30: 1/4 academic, 60% travel, rest other
• Premium users 4.2× as likely to finish as standard; 5% install premium; standard finish rate 3.8%

• a(i) Complete the probability tree. What is the probability a user is under 18 and uses the app for travel? Achieved
• a(ii) Out of 40 users, how many would you expect to be 18-30 using the app for academic or travel purposes? Merit
• a(iii) Given 3× as many over-30 users use it for travel as for other, and 55% of all users use it for travel overall, find the probability an over-30 user uses it for academic purposes. Excellence
• a(iv) Given course-completion rates of 10%/6%/2.5% by purpose for 18-30 year olds, find the probability a random 18-30 year old finishes the course. Merit
• b Given premium users are 4.2× as likely to finish as standard users (3.8%), and 5% of users are premium, find the overall probability of finishing the course. Merit

Question Three — Language, music, and cognitive decline

A longitudinal study of 964 older adults tracks whether they learned a language or instrument before 18, and whether they developed mild cognitive impairment (MCI).

• Language table: None 130/134, 1-4yrs 224/352, 4+yrs 42/82 (MCI/no MCI)
• Music table: 4+yrs 85/173; relative risk of MCI for 4+yrs music vs none = 0.708

• a What percentage of participants developed MCI, and what is the probability of MCI given they learned a language for at least 1 year? Achieved
• b Susie claims it's twice as likely to develop MCI learning music 4+ years vs language 4+ years, based on 85 vs 42. Explain her incorrect reasoning and correct her with calculations. Merit
• c(i) Given the relative risk of MCI for 4+ years of music vs none is 0.708, calculate the number of participants who developed MCI but learned no music. Merit
• c(ii) Evaluate a claim that learning music or language for 4+ years reduces MCI risk by about 30%, using relative risk calculations from both tables. Excellence

The 2023 paper ↗ Marking schedule ↗

Question One — A claw arcade game

A claw game drops the toy 15% of the time; otherwise the win probability starts at 1/10 and increases by 1 each unwon game (1/9, 1/8...), up to 4 attempts, or 2 losses.

• P(drop) = 0.15; win probability starts at 1/10 and rises if not won
• With friends: 70% of visits, win 10%; alone: win about 1/4

• a(i) Complete the probability tree, then find P(drops on 1st game, wins on 2nd). Achieved
• a(ii) Find the probability of winning on the 3rd game. Merit
• a(iii) Find the probability of winning within a maximum of 4 attempts. Excellence
• b(i) Given 70% of visits are with friends (win 10%) and 30% alone (win ~25%), find the overall probability of winning the claw game. Achieved
• b(ii) Given a treat is bought with different probabilities depending on how the game was won or lost, and the overall treat probability is 0.40, find the probability of friends, winning, and getting a treat. Excellence

Question Two — NZ media usage

1420 people surveyed on media usage by age group (youth 15-24 vs adults 25+); streaming-service usage tracked across four survey years.

• Online video: youth 228/251, adults 610/1169
• Streaming by year: 2016 102/249, 2020 181/262, 2021 191/251, 2022 473/706

• a(i) Find the probability a randomly selected person is a NZ youth who watched online video. Achieved
• a(ii) Compare the probability of watching online video given youth vs given adult, and evaluate a claim (Tao's) about which group watches more. Merit
• a(iii) Find the overall percentages watching online video, streaming services, and TV. Achieved
• a(iv) Calculate relative risks for streaming and for TV, youth vs adults, and evaluate two claims about how much more/less likely youth are to watch each. Excellence
• b(i) Find how much higher the percentage watching streaming services was in 2022 than 2016. Achieved
• b(ii) If 400 youth were surveyed in 2023, estimate how many would watch streaming services, justifying your estimate from the trend. Merit

Question Three — Normal distribution and screen time

NZ youth watch streaming (mean 113 min/day, σ = 35) and online video (mean 94 min/day, σ = 30); TV viewing data is skewed by non-watchers.

• Streaming: μ = 113 min, σ = 35 min
• Online video: μ = 94 min, σ = 30 min
• TV: overall mean 34 min; excluding non-watchers, mean 114 min; 12% of TV watchers exceed 150 min/day

• a(i) Find the probability a NZ youth spends under 92 minutes/day watching streaming services. Achieved
• a(ii) Find the probability a NZ youth spends between two and three hours daily watching streaming services. Merit
• b(i) Find the interquartile range for minutes spent watching online videos. Merit
• b(ii) Out of 50 youth, how many more would you expect to have spent over 120 minutes on streaming than on online video? Merit
• b(iii) Give at least TWO reasons why a normal distribution may not be appropriate for streaming or online-video minutes. Excellence
• c(i) Explain the effect on the mean and distribution shape of including youth who watch zero TV. Merit
• c(ii) Given mean 114 minutes (excluding non-watchers) and 12% watch over 150 minutes, calculate the standard deviation. Excellence

Grades above are read from each year's own Achievement/Merit/Excellence evidence statement, condensed to the actual task — the schedules also grade holistically per question on a 0–8 scale, not a fixed mark per bullet.

Past exams & resources

⏱ Timed opens the paper and starts a countdown matching the exam length. Official NZQA PDFs. If a link 404s, that year’s paper may sit under a slightly different code: browse the NZQA subject page.